How Can You Integrate Using Mathematica and Taylor Series?

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Integrating using Mathematica can be challenging, particularly for certain functions. A suggested approach is to compute the Taylor series of the function first. Once the series is obtained, it can be integrated term by term. This method allows for expressing the integral as an infinite series. Utilizing the Taylor series is often the most effective strategy when direct integration fails.
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How would you do this integral?
I have tried this using Mathematica, but failed to generate the answer.
Any suggestions?
 

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no can do...
 
The absolute best you can do it work out the taylor series and integrate term by term, and express it as an infinite series.
 

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